Homogeneous para-Kahler Einstein manifolds

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HOMOGENEOUS PARA-K ¨AHLER EINSTEIN MANIFOLDS D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI Dedicated to E.B.Vinberg on the occasion of his 70th birthday Abstract.A para-K¨a hler manifold can be defined as a pseudo-Riemannian manifold (M,g )with a parallel skew-symmetric para-complex structures K ,i.e.a parallel field of skew-symmetric endomor-phisms with K 2=Id or,equivalently,as a symplectic manifold (M,ω)with a bi-Lagrangian structure L ±,i.e.two complementary integrable Lagrangian distributions.A homogeneous manifold M =G/H of a semisimple Lie group G admits an invariant para-K¨a hler structure (g,K )if and only if it is a covering of the adjoint orbit Ad G h of a semisimple element h.We give a description of all invariant para-K¨a hler structures (g,K )on a such homogeneous ing a para-complex analogue of basic formulas of K¨a hler geometry,we prove that any invariant para-complex structure K on M =G/H defines a unique para-K¨a hler Einstein structure (g,K )with given non-zero scalar curvature.An explicit formula for the Einstein metric g is given.A survey of recent results on para-complex geometry is included.Contents 1.Introduction.22.A survey on para-complex geometry.42.1.Para-complex structures.42.2.Para-hypercomplex (complex product)structures.62.3.Para-quaternionic structures.72.4.Almost para-Hermitian and para-Hermitian structures.82.5.Para-K¨a hler (bi-Lagrangian or Lagrangian 2-web)structures.11

2.6.Para-hyperK¨a hler (hypersymplectic)structures and para-

hyperK¨a hler structures with torsion (PHKT-structures).13

2.7.Para-quaternionic K¨a hler structures and para-quaternionic-

K¨a hler with torsion (PQKT)structures.14

2 D.V.ALEKSEEVSKY,C.MEDORI AND A.TOMASSINI

2.8.Para-CR structures and para-quaternionic CR structures

(para-3-Sasakian structures).16 3.Para-complex vector spaces.17 3.1.The algebra of para-complex numbers C.17 3.2.Para-complex structures on a real vector space.18

3.3.Para-Hermitian forms.19

4.Para-complex manifolds.19 4.1.Para-holomorphic coordinates.20

4.2.Para-complex differential forms.21

5.Para-K¨a hler manifolds.22 5.1.Para-K¨a hler structures and para-K¨a hler potential.22 5.2.Curvature tensor of a para-K¨a hler metric.24 5.3.The Ricci form in para-holomorphic coordinates.25

5.4.The canonical form of a para-complex manifold.27

6.Homogeneous para-K¨a hler manifolds.29 6.1.The Koszul formula for the canonical form.29 6.2.Invariant para-complex structures on a homogeneous manifold.31 6.3.Invariant para-K¨a hler structures on a homogeneous reductive

manifold.32 7.Homogeneous para-K¨a hler Einstein manifolds of a semisimple

group.33 7.1.Invariant para-K¨a hler structures on a homogeneous manifold.33 7.2.Fundamental gradations of a real semisimple Lie algebra.34 putation of the Koszul form and the main theorem.36 7.4.Examples.38 References40

1.Introduction.

An almost para-complex structure on a2n-dimensional manifold M is afield K of involutive endomorphisms(K2=1)with n-dimensional eigendistributions T±with eigenvalues±1.(More generally,anyfield K of involutive endomorphisms is called an almost para-complex struc-ture in weak sense).

If the n-dimensional eigendistributions T±of K are involutive,thefield K is called a para-complex structure.This is equivalent to the vanishing of the Nijenhuis tensor N K of the K.In other words,a para-complex struc-ture on M is the same as a pair of complementary n-dimensional integrable distributions T±M.

A decomposition M=M+×M−of a manifold M into a direct product defines on M a para-complex structure K in the weak sense with eigendis-tributions T+=T M+and T−=T M−tangent to the factors.It is a para-complex structure in the factors M±have the same dimension.

Any para-complex structure locally can be obtained by this construction.